Srinivasa Ramanujan

For the algebraic geometer see C. P. Ramanujam.

Srinivasa Ramanujan
Born	December 22, 1887 Erode, Tamil Nadu, India
Died	April 26, 1920 Chetput, (Chennai), Tamil Nadu, India
Residence	India, UK
Nationality	Indian
Field	Mathematician
Alma mater	University of Cambridge
Academic advisor	G. H. Hardy and J. E. Littlewood
Known for	Landau-Ramanujan constant Ramanujan-Soldner constant Ramanujan theta function Rogers-Ramanujan identity Ramanujan prime Mock theta functions Ramanujan's sum
Religious stance	Hindu

Srinivasa Ramanujan Iyengar (Tamil: ஸ்ரீனிவாச ராமானுஜன்) (December 22, 1887 – April 26, 1920) was an Indian mathematician widely regarded as one of the greatest mathematicians in modern history. With almost no formal training in pure mathematics, he made substantial contributions in the areas of analysis, number theory, infinite series and continued fractions.

A definite example of an autodidact prodigy, Ramanujan compiled over 3000 theorems during his short lifetime. Although a small number of these results were actually false, most of his statements have now been proven to be correct.. His deep intuition and uncanny algebraic manipulative ability enabled him to state results that were both original and highly unconventional, and these have inspired a vast amount of new research. However, some of his major discoveries have been rather slow to enter the mathematical mainstream. Recently, Ramanujan's formulae have found applications in the field of crystallography and in string theory. The Ramanujan Journal, an international publication, was launched to publish work in all the areas of mathematics that were influenced by Ramanujan.

Life

Childhood and early life

Ramanujan was born in 1887 in Erode, Tamil Nadu, India, at the place of residence of his maternal grandparents. His father, K.Srinivasa Iyengar worked as an accountant and hailed from the fertile district of Thanjavur. His mother, Komalatammal was a housewife. They lived in Saarangapani Street in a south-Indian-style home (now a museum) in the town of Kumbakonam. In 1898, at the age of 10, Ramanujan entered the town high school, THSS, where he encountered formal mathematics for the first time. By the age of 11, he had exhausted the mathematical knowledge of two college students, who were lodgers at his home. He was later lent advanced trigonometry written by S.L. Loney (ISBN 1-4181-8509-4) and he completely mastered this book by the age of 13 and was discovering sophisticated theorems of his own. His biographer reports that by 14 his true genius was evident. He achieved merit certificates and academic awards throughout his school career and was also assisting the school in the logistics of assigning its 1200 students (each with their own needs) to its 35-odd teachers, completing mathematical exams in half the allotted time, and showing familiarity with infinite series. When he was sixteen and in the sixth form, he came across ``A synopsis of elementary results in pure and applied mathematics" (ISBN 0-8284-0239-6), by George Carr. This book, a collection of 6000 theorems, served to introduce Ramanujan to the real world of mathematics, but in a highly personal style that relegated the proofs, if any, to mere footnotes. By the age of 17, he had independently developed and investigated the Bernoulli numbers and had calculated Euler's constant up to 15 decimal places. His peers of the time commented later, "We, including teachers, rarely understood him" and "stood in respectful awe" of him.

Ramanujan received a scholarship to study at Government College in Kumbakonam but was so intent on studying mathematics that he could not focus on any other subjects and failed most of them, losing his scholarship in the process. He failed again in the next college he joined but continued to pursue independent research in mathematics. At this point in his life, he lived in extreme poverty and was often near the point of starvation.

Adulthood in India

In 1909, Ramanujan was married to a nine-year old bride, Janaki Ammal, as per the customs of India at that time and began searching for a job. With his collection of mathematical results, he travelled door to door around the city of Madras (now Chennai) looking for a clerical position. Eventually, he found a position in the accountant general's office and subsequently in the accounts section of the Madras Port Trust. Ramanujan wanted to focus his time completely on mathematics and needed financial help to carry on his research. He solicited support from many influential Indians and published several papers in Indian mathematical journals, but was unsuccessful in his attempts to foster sponsorship. It might be the case that he was supported by Ramachandra Rao, then the collector of the Nellore district and a distinguished civil servant. Rao, an amateur mathematician himself, was the uncle of the well-known mathematician, K. Ananda Rao, who went on to become the Principal of the Presidency College.

Following his supervisor's advice, Ramanujan, in late 1912 and early 1913, sent letters and samples of his theorems to three Cambridge academics: H. F. Baker, E. W. Hobson, and G. H. Hardy. The first two professors returned his letters without any comments. On the other hand, Hardy had the foresight to quickly recognize Ramanujan as a genius. Upon reading the initial unsolicited missive by an unknown and untrained Indian mathematician, G.H. Hardy and his colleague J.E. Littlewood after discussion, concluded, “not one [theorem] could have been set in the most advanced mathematical examination in the world.” Although Hardy was one of the most eminent mathematicians of his day and an expert in a number of fields that Ramanujan was writing about, he commented that, "many of them [theorems] defeated me completely; I had never seen anything in the least like them before."

Life in England

After some initial scepticism, Hardy replied with comments, requesting proofs for some of the discoveries, and began to make plans to bring Ramanujan to Cambridge. Ramanujan was at first apprehensive to travel overseas due to religious reasons, but eventually his well-wishers prevailed upon him and he agreed to come to England. He spent nearly five years in Cambridge collaborating with Hardy and Littlewood and published a part of his findings there. Hardy and Ramanujan had highly contrasting personalities. Their collaboration was a clash of different cultures, beliefs and working styles. Hardy was an atheist and an apostle of proof and mathematical rigour, whereas, Ramanujan was a deeply religious man and relied very strongly on his intuition. While in England, Hardy tried his best to fill the gaps in Ramanujan's education without interrupting his spell of inspiration.

Ramanujan was awarded a B.A. degree in March 1916 for his work on Highly composite numbers which was published as a paper in the Journal of the London Mathematical Society. He was the second Indian to become a Fellow of the Royal Society in 1918 and he became one of the youngest Fellows in the entire history of the Royal Society. He was elected “for his investigation in Elliptic Functions and the Theory of Numbers.” On 1918-10-13 he became the first Indian to be elected a Fellow of Trinity College, Cambridge.

Illness and return to India

Plagued by health problems all through his life, living in a country far away from home, and obsessively involved with his mathematics, Ramanujan's health worsened in England, perhaps exacerbated by stress, and by the scarcity of vegetarian food during the First World War. He was diagnosed with tuberculosis (Henderson, 1996) and a severe vitamin deficiency and was confined to a sanatorium. Ramanujan returned to Kumbakonam, India in 1919 and died soon thereafter at the age of 32. His wife, S. Janaki Ammal, lived in Chennai (formerly Madras) until her death in 1994.

A 1994 analysis of Ramanujan's medical records and symptoms by Dr. D.A.B. Young concluded that it was much more likely he had hepatic amoebiasis, a parasitic infection of the liver. This is supported by the fact that Ramanujan had spent time in Madras, where the disease was widespread. He had had two cases of dysentery before he left India. When not properly treated, dysentery can lie dormant for years and lead to hepatic amoebiasis. It was a difficult disease to diagnose, but once diagnosed would have been readily curable (Berndt, 1998).

Personality

Ramanujan has been described as a person with a somewhat shy and quiet disposition, a dignified man. He was also known to be extremely sensitive. On one occasion, he had prepared a buffet for a number of guests, and when one guest politely refused to taste a dish he had prepared, he left immediately and took a taxi to Oxford. He also lived a rather spartan life while at Cambridge. He frequently cooked vegetables alone in his room.

Spiritual life

Ramanujan believed in Hindu gods all his life and lived as an observant Tamil Brahmin. "Iyengar" refers to a class of brahmins in southern India who worship the god Vishnu, the preserver of the universe. His first Indian biographers describe him as rigorously orthodox. Ramanujan credited his acumen to his family goddess, Namagiri, and looked to her for inspiration in his work. He often said, "An equation for me has no meaning, unless it represents a thought of God."

Mathematical achievements

In mathematics, there is a distinction between having an insight and having a proof. Ramanujan's talent suggested a plethora of formulae that could then be investigated in depth later. It is said that Ramanujan's discoveries are unusually rich and that there is often more in it than what initially meets the eye. As a by-product, new directions of research were opened up. Examples of the most interesting of these formulas include the intriguing infinite series for π, one of which is given below

${\frac {1}{\pi }}={\frac {2{\sqrt {2}}}{9801}}\sum _{k=0}^{\infty }{\frac {(4k)!(1103+26390k)}{(k!)^{4}396^{4k}}}$

This result is based on the negative fundamental discriminant d = –4×58 with class number h(d) = 2 (note that 57×13×58 = 26390) and is related to the fact that,

$e^{\pi {\sqrt {58}}}=396^{4}-104.000000177\dots .$

Ramanujan's series for π converges extraordinarily rapidly (exponentially) and forms the basis of some of the fastest algorithms currently used to calculate π.

His intuition also led him to derive some previously unknown identities, such as

$\left[1+2\sum _{n=1}^{\infty }{\frac {\cos(n\theta )}{\cosh(n\pi )}}\right]^{-2}+\left[1+2\sum _{n=1}^{\infty }{\frac {\cosh(n\theta )}{\cosh(n\pi )}}\right]^{-2}={\frac {2\Gamma ^{4}\left({\frac {3}{4}}\right)}{\pi }}$

for all $\theta$, where $\Gamma (z)$ is the gamma function. Equating coefficients of $\theta ^{0}$, $\theta ^{4}$, and $\theta ^{8}$ gives some deep identities for the hyperbolic secant.

In 1918, G. H. Hardy and Ramanujan studied the partition function P(n) extensively and gave a very accurate non-convergent asymptotic series for the number of partitions of an integer. Hans Rademacher, in 1937, was able to refine their formula to find an exact convergent series solution to this problem. This astonishing non-intuitive formula was a spectacular achievement in analytical number theory. Ramanujan and Hardy's work in this area gave rise to a powerful new method called the circle method which has found tremendous applications.

The Ramanujan conjecture

Although there are numerous statements that could bear the name Ramanujan conjecture, there is one in particular that was very influential on later work. In particular, the connection of that conjecture with conjectures of A.Weil in algebraic geometry opened new areas of research. That Ramanujan conjecture is an assertion on the size of the tau function, which has as generating function the discriminant modular form Δ(q), a typical cusp form in the theory of modular forms. It was finally proved in 1973, as a consequence of Pierre Deligne's proof of the Weil conjectures. The reduction step involved is complicated. Deligne won a Fields Medal for his work on Weil conjectures.

Ramanujan's notebooks

While he was still in India, Ramanujan recorded the bulk of his results in four notebooks of loose leaf paper. These results were mostly written up without any derivations. This is probably the origin of the misperception that Ramanujan was unable to prove his results and simply thought up the final result directly. Mathematician Bruce Berndt, in his review of these notebooks and Ramanujan's work, says that Ramanujan most certainly was able to make the proofs of most of his results, but chose not to.

This style of working may have been for several reasons. Since paper was very expensive, Ramanujan would do most of his work and perhaps his proofs on slate, and then transfer just the results to paper. Using a slate was common for mathematics students in India at the time. He was also quite likely to have been influenced by the style of one of the books from which he had learned much of his advanced mathematics: G. S. Carr's Synopsis of Pure and Applied Mathematics (ISBN 0-8284-0239-6), used by Carr in his tutoring. It summarised several thousand results, stating them without proofs. Finally, it is possible that Ramanujan considered his workings to be for his personal interest alone; and therefore only recorded the results. (Berndt, 1998)

The first notebook has 351 pages with 16 somewhat organized chapters and some unorganized material. The second notebook has 256 pages in 21 chapters and 100 unorganized pages, with the third notebook containing 33 unorganized pages. The results in his notebooks inspired numerous papers by later mathematicians trying to prove what he had found. Hardy himself created papers exploring material from Ramanujan's work as did G. N. Watson, B. M. Wilson, and Bruce Berndt. (Berndt, 1998) A fourth notebook, the so-called "lost notebook", was rediscovered in 1976 by George Andrews.

Other mathematicians' views of Ramanujan

Ramanujan is generally hailed as an all time great mathematician like Euler, Gauss or Jacobi for his natural genius⁵. G. H. Hardy quotes: "The limitations of his knowledge were as startling as its profundity. Here was a man who could work out modular equations and theorems... to orders unheard of, whose mastery of continued fractions was... beyond that of any mathematician in the world, who had found for himself the functional equation of the zeta function and the dominant terms of many of the most famous problems in the analytic theory of numbers; and yet he had never heard of a doubly-periodic function or of Cauchy's theorem, and had indeed but the vaguest idea of what a function of a complex variable was..."

Quoting K. Srinivasa Rao"As for his place in the world of Mathematics, we quote Bruce C Berndt: 'Paul Erdős has passed on to us G. H. Hardy's personal ratings of mathematicians. Suppose that we rate mathematicians on the basis of pure talent on a scale from 0 to 100, Hardy gave himself a score of 25, J.E. Littlewood 30, Hilbert 80 and Ramanujan 100.'"

In his book The Scientific Edge, noted physicist Jayant Narlikar says that “Srinivasa Ramanujan, discovered by the Cambridge mathematician G.H. Hardy, whose great mathematical findings were beginning to be appreciated from 1915 to 1919. His achievements were to be fully understood much later, well after his untimely death in 1920. For example, his work on the highly composite numbers (numbers with a large number of factors) started a whole new line of investigations in the theory of such numbers.” Narlikar also goes on to say that his work was one of the top ten achievements of 20^th century Indian science and “could be considered in the Nobel Prize class.” (Narlikar 2003 p127) The work of other 20^th century Indian scientists which Narlikar considered to be of Nobel Prize class were those of Chandrasekhara Venkata Raman, Meghnad Saha and Satyendra Nath Bose.

Recognition

Ramanujan's home state of Tamil Nadu celebrates December 22 (Ramanujan's birthday) as 'State IT Day', memorializing both the man and his achievements, as a native of Tamil Nadu.

A stamp picturing Ramanujan was released by the Government of India in 1962 — the 75^th anniversary of Ramanujan's birth — commemorating his achievements in the field of number theory.

A prize for young mathematicians from developing countries has been created in the name of Srinivasa Ramanujan by the International Centre for Theoretical Physics (ICTP), in cooperation with the IMU, who nominate members of the prize committee.

During the year 1987 (Ramanujan's centennial), the printed form of Ramanujan's Lost Notebook by Springer-Narosa was released by the late prime minister, Rajiv Gandhi, who presented the first copy to S. Janaki Ammal Ramanujan (Ramanujan's late widow) and the second copy to George Andrews in recognition of his contributions in the field of number theory.

Projected films

• An international feature film on Ramanujan's life will begin shooting in 2007 in Tamil Nadu state and Cambridge. It is being produced by an Indo-British collaboration; it will be co-directed by Stephen Fry and Dev Benegal. A play First Class Man by Alter Ego Productions was based on David Freeman's "First Class Man." The play is centered around Ramanujan and his complex and dysfunctional relationship with G. H. Hardy.
• Another film based on the book The Man Who Knew Infinity: A Life of the Genius Ramanujan by Robert Kanigel is being made by Edward Pressman and Matthew Brown.

Cultural references

• He was referred to in the film Good Will Hunting as an example of mathematical genius.
• His biography was highlighted in the Vernor Vinge book The Peace War as well as Douglas Hofstadter's Gödel, Escher, Bach.
• The character "Amita Ramanujan" in the CBS TV series Numb3rs (2005-) was named after him (source: IMDB's trivia for 'Numb3rs').
• The short story "Gomez," by Cyril Kornbluth, mentions Ramanujan by name as a comparison to its title character, another self-taught mathematical genius.
• In the novel Uncle Petros and Goldbach's Conjecture by Apostolos Doxiadis, Ramanujan is one of the characters.
• In the novel Earth by David Brin, the character Jen Wolling uses a representation of Sri Ramanujan as her computer interface.
• In the novel The Peace War by Vernor Vinge, a young mathematical genius is referred to as "my little Ramanujan" accidentally. Then it is hoped the young man doesn't get the connection because, like Ramanujan, the boy is doomed to die prematurely.
• The character "Yugo Amaryl" in Isaac Asimov's Prelude to Foundation is based on Ramanujan.
• The theatre company Complicite have created a production based around the life of Ramanjuan called A Disappearing Number - conceived and directed by Simon McBurney

Further reading

• Kanigel, Robert. The Man Who Knew Infinity: A Life of the Genius Ramanujan (Washington Square Press, 1991) ISBN 0-671-75061-5
• Narlikar, Jayant V. The Scientific Edge (Penguin Books, 2003)
• Peterson, Doug. "Raiders of the Lost Notebook: LAS mathematician tracks proof for legendary numbers genius.", University of Illinois LASNews, Fall/Winter 2006.

See also

• List of topics named after Srinivasa Ramanujan
• Ramanujan-Peterssen conjecture
• 1729 (number)
• Landau-Ramanujan constant
• Ramanujan-Soldner constant
• Ramanujan summation
• Ramanujan theta function
• Ramanujan graph
• Ramanujan's tau function
• Rogers-Ramanujan identity
• Ramanujan prime
• Ramanujan's constant (hoax)
• Ramanujan's sum
• Ramanujan modular functions

Selected publications by Ramanujan

• Collected Papers of Srinivasa Ramanujan, by Srinivasa Ramanujan, G. H. Hardy, P. V. Seshu Aiyar, B. M. Wilson, Bruce C. Berndt (AMS, 2000, ISBN 0-8218-2076-1)

This book was originally published in 1927 after Ramanujan's death. It contains the 37 papers published in professional journals by Ramanujan during his lifetime. The third re-print contains additional commentary by Bruce C. Berndt.

• Notebooks (2 Volumes), S. Ramanujan, Tata Institute of Fundamental Research, Bombay, 1957.

These books contain photo copies of the original notebooks as written by Ramanujan.

• The Lost Notebook and Other Unpublished Papers, by S. Ramanujan, Narosa, New Delhi, 1988.

This book contains photo copies of the pages in the "Lost Notebook."

Selected publications about Ramanujan or his work

• Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work by G. H. Hardy (AMS, 1999, ISBN 0-8218-2023-0)
• Ramanujan: Letters and Commentary (History of Mathematics, V. 9), by Bruce C. Berndt and Robert A. Rankin (American Mathematical Society, 1995, ISBN 0-8218-0287-9)
• Ramanujan: Essays and Surveys (History of Mathematics, V. 22), by Bruce C. Berndt and Robert A. Rankin (American Mathematical Society, 2001, ISBN 0-8218-2624-7)
• Ramanujan's Notebooks, Part I, by Bruce C. Berndt (Springer, 1985, ISBN 0-387-96110-0)
• Ramanujan's Notebooks, Part II, by Bruce C. Berndt (Springer, 1999, ISBN 0-387-96794-X)
• Ramanujan's Notebooks, Part III, by Bruce C. Berndt (Springer, 2004, ISBN 0-387-97503-9)
• Ramanujan's Notebooks, Part IV, by Bruce C. Berndt (Springer, 1993, ISBN 0-387-94109-6)
• Ramanujan's Notebooks, Part V, by Bruce C. Berndt (Springer, 2005, ISBN 0-387-94941-0)
• Ramanujan's Lost Notebook, Part I, by George Andrews and Bruce C. Berndt (Springer, 2005, ISBN 0-387-25529-X)
• Number Theory in the Spirit of Ramanujan by Bruce C. Berndt (AMS, 2006, ISBN 0-8218-4178-5)
• An overview of Ramanujan's notebooks by Bruce C. Berndt, in Charlemagne and His Heritage: 1200 Years of Civilization and Science in Europe, Volume 2: Mathematical Arts, P. L. Butzer, H. Th. Jongen, and W. Oberschelp, editors, Brepols, Turnhout, 1998, pp. 119-146, (22 pg. pdf file)
• Modern Mathematicians, Harry Henderson, (Facts on File Inc., 1996, ISBN 0-8160-3235-1)
• The Man Who Knew Infinity: A Life of the Genius Ramanujan by Robert Kanigel (1991, ISBN 0-671-75061-5)

External links

• Film to celebrate mathematics genius. BBC. Retrieved August 24, 2006.
• O'Connor, John J.; Edmund F. Robertson "Srinivasa Ramanujan". MacTutor History of Mathematics archive.
• Eric W. Weisstein, Ramanujan, Srinivasa (1887-1920) at ScienceWorld.
• Biographical essay on Ramanujan
• Biography of this mathematical genius at World of Biography
• A Study Group For Mathematics: Srinivasa Ramanujan Iyengar
• Srinivasan Ramanujan in One Hundred Tamils of 20th Century
• The Ramanujan Journal - An international journal devoted to Ramanujan
• Jai Maharaj, Computing the Mathematical Face of God: S. Ramanujan
• Srinivasa Aiyangar Ramanujan
• A short biography of Ramanujan
• International Math Union Prizes, including a Ramanujan Prize.
• A biographical song about Ramanujan's life
• Feature Film on Math Genius Ramanujan by Dev Benegal and Stephen Fry
• A magical genius
• Norwegian and Indian mathematical geniuses
• RAMANUJAN — Essays and Surveys
• Ramanujan's growing influence
• Ramanujan's mentor
• A biography of Ramanujan set to music
• "A passion for numbers"
• BBC radio programme about Ramanujan - episode 5
• Bruce C. Berndt, Robert A. Rankin. The Books Studied by Ramanujan in India American Mathematical Monthly, Vol. 107, No. 7 (Aug. - Sep., 2000), pp. 595-601 doi:10.2307/2589114
• "Ramanujan's mock theta function puzzle solved"
• Example of an original page from the second notebook [1]

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